References & Citations

Bookmark

Quantum Physics

Title:
Quantum Knots and Mosaics

Abstract: In this paper, we give a precise and workable definition of a quantum knot
system, the states of which are called quantum knots. This definition can be
viewed as a blueprint for the construction of an actual physical quantum
system.
Moreover, this definition of a quantum knot system is intended to represent
the "quantum embodiment" of a closed knotted physical piece of rope. A quantum
knot, as a state of this system, represents the state of such a knotted closed
piece of rope, i.e., the particular spatial configuration of the knot tied in
the rope. Associated with a quantum knot system is a group of unitary
transformations, called the ambient group, which represents all possible ways
of moving the rope around (without cutting the rope, and without letting the
rope pass through itself.)
Of course, unlike a classical closed piece of rope, a quantum knot can
exhibit non-classical behavior, such as quantum superposition and quantum
entanglement. This raises some interesting and puzzling questions about the
relation between topological and quantum entanglement.
The knot type of a quantum knot is simply the orbit of the quantum knot under
the action of the ambient group. We investigate quantum observables which are
invariants of quantum knot type. We also study the Hamiltonians associated with
the generators of the ambient group, and briefly look at the quantum tunneling
of overcrossings into undercrossings.
A basic building block in this paper is a mosaic system which is a formal
(rewriting) system of symbol strings. We conjecture that this formal system
fully captures in an axiomatic way all of the properties of tame knot theory.